LMFDB - Embedded newform 297.2.e.b.100.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [297,2,Mod(100,297)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("297.100"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(297, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 297 = 3^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	297.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.37155694003$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			100.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	297.100
Dual form			297.2.e.b.199.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.73205i) q^{4} +(-1.50000 - 2.59808i) q^{5} +(2.00000 - 3.46410i) q^{7} +(0.500000 - 0.866025i) q^{11} +(-1.00000 - 1.73205i) q^{13} +(-2.00000 + 3.46410i) q^{16} +6.00000 q^{17} +2.00000 q^{19} +(3.00000 - 5.19615i) q^{20} +(1.50000 + 2.59808i) q^{23} +(-2.00000 + 3.46410i) q^{25} +8.00000 q^{28} +(-3.00000 + 5.19615i) q^{29} +(-4.00000 - 6.92820i) q^{31} -12.0000 q^{35} +2.00000 q^{37} +(-4.00000 + 6.92820i) q^{43} +2.00000 q^{44} +(1.50000 - 2.59808i) q^{47} +(-4.50000 - 7.79423i) q^{49} +(2.00000 - 3.46410i) q^{52} -3.00000 q^{53} -3.00000 q^{55} +(-4.00000 + 6.92820i) q^{61} -8.00000 q^{64} +(-3.00000 + 5.19615i) q^{65} +(6.50000 + 11.2583i) q^{67} +(6.00000 + 10.3923i) q^{68} +2.00000 q^{73} +(2.00000 + 3.46410i) q^{76} +(-2.00000 - 3.46410i) q^{77} +(-1.00000 + 1.73205i) q^{79} +12.0000 q^{80} +(-9.00000 + 15.5885i) q^{83} +(-9.00000 - 15.5885i) q^{85} -3.00000 q^{89} -8.00000 q^{91} +(-3.00000 + 5.19615i) q^{92} +(-3.00000 - 5.19615i) q^{95} +(-1.00000 + 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 3 q^{5} + 4 q^{7} + q^{11} - 2 q^{13} - 4 q^{16} + 12 q^{17} + 4 q^{19} + 6 q^{20} + 3 q^{23} - 4 q^{25} + 16 q^{28} - 6 q^{29} - 8 q^{31} - 24 q^{35} + 4 q^{37} - 8 q^{43} + 4 q^{44} + 3 q^{47}+ \cdots - 2 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 - 3 * q^5 + 4 * q^7 + q^11 - 2 * q^13 - 4 * q^16 + 12 * q^17 + 4 * q^19 + 6 * q^20 + 3 * q^23 - 4 * q^25 + 16 * q^28 - 6 * q^29 - 8 * q^31 - 24 * q^35 + 4 * q^37 - 8 * q^43 + 4 * q^44 + 3 * q^47 - 9 * q^49 + 4 * q^52 - 6 * q^53 - 6 * q^55 - 8 * q^61 - 16 * q^64 - 6 * q^65 + 13 * q^67 + 12 * q^68 + 4 * q^73 + 4 * q^76 - 4 * q^77 - 2 * q^79 + 24 * q^80 - 18 * q^83 - 18 * q^85 - 6 * q^89 - 16 * q^91 - 6 * q^92 - 6 * q^95 - 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/297\mathbb{Z}\right)^\times$.

$n$	$56$	$244$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$. You can download additional coefficients here.

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$2$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$3$		0			0
$3$		0			0
$4$	1.00000	+	1.73205i	0.500000	+	0.866025i
$5$	−1.50000	−	2.59808i	−0.670820	−	1.16190i	−0.977672	−	0.210138i	$-0.932609\pi$
$5$	−1.50000	−	2.59808i	−0.670820	−	1.16190i	0.306851	−	0.951757i	$-0.400725\pi$
$6$		0			0
$7$	2.00000	−	3.46410i	0.755929	−	1.30931i	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	2.00000	−	3.46410i	0.755929	−	1.30931i	0.944911	−	0.327327i	$-0.106148\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.500000	−	0.866025i	0.150756	−	0.261116i
$11$	0.500000	−	0.866025i	0.150756	−	0.261116i
$12$		0			0
$13$	−1.00000	−	1.73205i	−0.277350	−	0.480384i	0.693375	−	0.720577i	$-0.256123\pi$
$13$	−1.00000	−	1.73205i	−0.277350	−	0.480384i	−0.970725	+	0.240192i	$0.922790\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$	6.00000			1.45521			0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000			1.45521			0.727607	+	0.685994i	$0.240633\pi$
$18$		0			0
$19$	2.00000			0.458831			0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000			0.458831			0.229416	+	0.973329i	$0.426318\pi$
$20$	3.00000	−	5.19615i	0.670820	−	1.16190i
$21$		0			0
$22$		0			0
$23$	1.50000	+	2.59808i	0.312772	+	0.541736i	0.978961	−	0.204046i	$-0.0654092\pi$
$23$	1.50000	+	2.59808i	0.312772	+	0.541736i	−0.666190	+	0.745782i	$0.732076\pi$
$24$		0			0
$25$	−2.00000	+	3.46410i	−0.400000	+	0.692820i
$26$		0			0
$27$		0			0
$28$	8.00000			1.51186
$29$	−3.00000	+	5.19615i	−0.557086	+	0.964901i	0.440652	+	0.897678i	$0.354747\pi$
$29$	−3.00000	+	5.19615i	−0.557086	+	0.964901i	−0.997738	+	0.0672232i	$0.978586\pi$
$30$		0			0
$31$	−4.00000	−	6.92820i	−0.718421	−	1.24434i	−0.961625	−	0.274367i	$-0.911532\pi$
$31$	−4.00000	−	6.92820i	−0.718421	−	1.24434i	0.243204	−	0.969975i	$-0.421802\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−12.0000			−2.02837
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$41$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$42$		0			0
$43$	−4.00000	+	6.92820i	−0.609994	+	1.05654i	0.381246	+	0.924473i	$0.375495\pi$
$43$	−4.00000	+	6.92820i	−0.609994	+	1.05654i	−0.991241	+	0.132068i	$0.957838\pi$
$44$	2.00000			0.301511
$45$		0			0
$46$		0			0
$47$	1.50000	−	2.59808i	0.218797	−	0.378968i	−0.735643	−	0.677369i	$-0.763120\pi$
$47$	1.50000	−	2.59808i	0.218797	−	0.378968i	0.954441	+	0.298401i	$0.0964533\pi$
$48$		0			0
$49$	−4.50000	−	7.79423i	−0.642857	−	1.11346i
$50$		0			0
$51$		0			0
$52$	2.00000	−	3.46410i	0.277350	−	0.480384i
$53$	−3.00000			−0.412082			−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000			−0.412082			−0.206041	+	0.978543i	$0.566058\pi$
$54$		0			0
$55$	−3.00000			−0.404520
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	−4.00000	+	6.92820i	−0.512148	+	0.887066i	0.487753	+	0.872982i	$0.337817\pi$
$61$	−4.00000	+	6.92820i	−0.512148	+	0.887066i	−0.999901	+	0.0140840i	$0.995517\pi$
$62$		0			0
$63$		0			0
$64$	−8.00000			−1.00000
$65$	−3.00000	+	5.19615i	−0.372104	+	0.644503i
$66$		0			0
$67$	6.50000	+	11.2583i	0.794101	+	1.37542i	0.923408	+	0.383819i	$0.125391\pi$
$67$	6.50000	+	11.2583i	0.794101	+	1.37542i	−0.129307	+	0.991605i	$0.541275\pi$
$68$	6.00000	+	10.3923i	0.727607	+	1.26025i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	2.00000			0.234082			0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000			0.234082			0.117041	+	0.993127i	$0.462659\pi$
$74$		0			0
$75$		0			0
$76$	2.00000	+	3.46410i	0.229416	+	0.397360i
$77$	−2.00000	−	3.46410i	−0.227921	−	0.394771i
$78$		0			0
$79$	−1.00000	+	1.73205i	−0.112509	+	0.194871i	−0.916781	−	0.399390i	$-0.869222\pi$
$79$	−1.00000	+	1.73205i	−0.112509	+	0.194871i	0.804272	+	0.594261i	$0.202555\pi$
$80$	12.0000			1.34164
$81$		0			0
$82$		0			0
$83$	−9.00000	+	15.5885i	−0.987878	+	1.71106i	−0.359506	+	0.933143i	$0.617055\pi$
$83$	−9.00000	+	15.5885i	−0.987878	+	1.71106i	−0.628372	+	0.777913i	$0.716279\pi$
$84$		0			0
$85$	−9.00000	−	15.5885i	−0.976187	−	1.69081i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.00000			−0.317999			−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000			−0.317999			−0.159000	+	0.987279i	$0.550827\pi$
$90$		0			0
$91$	−8.00000			−0.838628
$92$	−3.00000	+	5.19615i	−0.312772	+	0.541736i
$93$		0			0
$94$		0			0
$95$	−3.00000	−	5.19615i	−0.307794	−	0.533114i
$96$		0			0
$97$	−1.00000	+	1.73205i	−0.101535	+	0.175863i	−0.912317	−	0.409484i	$-0.865709\pi$
$97$	−1.00000	+	1.73205i	−0.101535	+	0.175863i	0.810782	+	0.585348i	$0.199042\pi$
$98$		0			0
$99$		0			0

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	297.2.e.b.100.1		2
3.2	odd	2		99.2.e.b.34.1	✓	2
9.2	odd	6		891.2.a.d.1.1		1
9.4	even	3	inner	297.2.e.b.199.1		2
9.5	odd	6		99.2.e.b.67.1	yes	2
9.7	even	3		891.2.a.e.1.1		1
33.32	even	2		1089.2.e.b.727.1		2
99.32	even	6		1089.2.e.b.364.1		2
99.43	odd	6		9801.2.a.g.1.1		1
99.65	even	6		9801.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.e.b.34.1	✓	2	3.2	odd	2
99.2.e.b.67.1	yes	2	9.5	odd	6
297.2.e.b.100.1		2	1.1	even	1	trivial
297.2.e.b.199.1		2	9.4	even	3	inner
891.2.a.d.1.1		1	9.2	odd	6
891.2.a.e.1.1		1	9.7	even	3
1089.2.e.b.364.1		2	99.32	even	6
1089.2.e.b.727.1		2	33.32	even	2
9801.2.a.f.1.1		1	99.65	even	6
9801.2.a.g.1.1		1	99.43	odd	6

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 297 = 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	297.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{4} +(-1.50000 - 2.59808i) q^{5} +(2.00000 - 3.46410i) q^{7} +(0.500000 - 0.866025i) q^{11} +(-1.00000 - 1.73205i) q^{13} +(-2.00000 + 3.46410i) q^{16} +6.00000 q^{17} +2.00000 q^{19} +(3.00000 - 5.19615i) q^{20} +(1.50000 + 2.59808i) q^{23} +(-2.00000 + 3.46410i) q^{25} +8.00000 q^{28} +(-3.00000 + 5.19615i) q^{29} +(-4.00000 - 6.92820i) q^{31} -12.0000 q^{35} +2.00000 q^{37} +(-4.00000 + 6.92820i) q^{43} +2.00000 q^{44} +(1.50000 - 2.59808i) q^{47} +(-4.50000 - 7.79423i) q^{49} +(2.00000 - 3.46410i) q^{52} -3.00000 q^{53} -3.00000 q^{55} +(-4.00000 + 6.92820i) q^{61} -8.00000 q^{64} +(-3.00000 + 5.19615i) q^{65} +(6.50000 + 11.2583i) q^{67} +(6.00000 + 10.3923i) q^{68} +2.00000 q^{73} +(2.00000 + 3.46410i) q^{76} +(-2.00000 - 3.46410i) q^{77} +(-1.00000 + 1.73205i) q^{79} +12.0000 q^{80} +(-9.00000 + 15.5885i) q^{83} +(-9.00000 - 15.5885i) q^{85} -3.00000 q^{89} -8.00000 q^{91} +(-3.00000 + 5.19615i) q^{92} +(-3.00000 - 5.19615i) q^{95} +(-1.00000 + 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 3 q^{5} + 4 q^{7} + q^{11} - 2 q^{13} - 4 q^{16} + 12 q^{17} + 4 q^{19} + 6 q^{20} + 3 q^{23} - 4 q^{25} + 16 q^{28} - 6 q^{29} - 8 q^{31} - 24 q^{35} + 4 q^{37} - 8 q^{43} + 4 q^{44} + 3 q^{47}+ \cdots - 2 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 - 3 * q^5 + 4 * q^7 + q^11 - 2 * q^13 - 4 * q^16 + 12 * q^17 + 4 * q^19 + 6 * q^20 + 3 * q^23 - 4 * q^25 + 16 * q^28 - 6 * q^29 - 8 * q^31 - 24 * q^35 + 4 * q^37 - 8 * q^43 + 4 * q^44 + 3 * q^47 - 9 * q^49 + 4 * q^52 - 6 * q^53 - 6 * q^55 - 8 * q^61 - 16 * q^64 - 6 * q^65 + 13 * q^67 + 12 * q^68 + 4 * q^73 + 4 * q^76 - 4 * q^77 - 2 * q^79 + 24 * q^80 - 18 * q^83 - 18 * q^85 - 6 * q^89 - 16 * q^91 - 6 * q^92 - 6 * q^95 - 2 * q^97

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